00000241.htm

224 IX ANALYTIC FUNCTIONS

Example

(9.8.4) Let s_n be the circuit / -»e"^lt defined in I = [0, 2n], n being a positive
or negative integer; we have e_n(l) = U; &_n is called "the unit circle taken
n times". We observe that the open set C - U has two connected components,
namely the ball B: |z| < 1 and the exterior E of B defined by \z\ > 1. Indeed,
B is connected as a star-shaped domain (9.7.1); and by Section 4.4 and
(9.5.7) E is the image of ]l,+oo[ x [0, 2n] by the continuous mapping
(x₉t)-*xJ*₉ hence the result by (3.19.1), (3.20.16), and (3.19.7) (a similar
argument also proves the connectedness of B and of B—{0}); finally in
C — U, B and E are open and closed since B is open in C and B = (C — U) n B,
and we have B n E = 0. From the definition and (9.5.3) it follows that
y(0; s_tt) = /?, hence yfo £„) = nfor any point z ofE. Let us show that j(z; s_n) = 0
for any point of E; more generally:

(9.8.5) If a circuit y is contained in a closed ball D: \z — a\ ^ r, thenj(z; y) = 0
for any point z exterior to D.

Indeed, suppose y is defined in an interval I = [b, c], and that |/(OI ^ M
in that interval. By definition,

"_f ^ ( ^dx f y(*)^tf*

y(z;y) = -----= —

J_yx-z Jby(i)-

for |z - a\ > r.

But as \y(t) - a\ < r, we have |y(f) — z\^\z — a\—r for any /el, and
therefore, by the mean value theorem,

M(c-fe)

^|z-a|-r'

when |z - a| is large enough the right-hand side is < 2n, and as j(z; y) is
an integer, this implies y(^ 5 ?) = 0. But the exterior of D is connected, as
seen above, hence the conclusion by (9.8.3).

(9.8.6) For any circuit y in C, defined in I, the set of points x e C - y(I)
such thatj(x; y) ^ 0 is relatively compact in C.

For by (9.8.5), that set is contained in any closed ball containing y(I).

(9.8.7) Let Aa C be a simply connected domain, y a circuit in A. For any
point x ofC - A,y(*; y) = 0.



224 IX ANALYTIC FUNCTIONS Example (9.8.4) Let s_n be the circuit / -»e"^lt defined in I = [0, 2n], n being a positive or negative integer; we have e_n(l) = U; &_n is called "the unit circle taken n times". We observe that the open set C - U has two connected components, namely the ball B: \|z\| < 1 and the exterior E of B defined by \z\ > 1. Indeed, B is connected as a star-shaped domain (9.7.1); and by Section 4.4 and (9.5.7) E is the image of ]l,+oo[ x [0, 2n] by the continuous mapping (x₉t)-xJ₉ hence the result by (3.19.1), (3.20.16), and (3.19.7) (a similar argument also proves the connectedness of B and of B—{0}); finally in C — U, B and E are open and closed since B is open in C and B = (C — U) n B, and we have B n E = 0. From the definition and (9.5.3) it follows that y(0; s_tt) = /?, hence yfo £„) = nfor any point z ofE. Let us show that j(z; s_n) = 0 for any point of E; more generally:

(9.8.5) If a circuit y is contained in a closed ball D: \z — a\ ^ r, thenj(z; y) = 0 for any point z exterior to D. Indeed, suppose y is defined in an interval I = [b, c], and that \|/(OI ^ M in that interval. By definition,

2m	"_f ^ ( ^dx f y()^tf y(z;y) = -----= — J_yx-z Jby(i)-	for \|z - a\ > r.

But as \y(t) - a\ < r, we have \|y(f) — z\^\z — a\—r for any /el, and therefore, by the mean value theorem, M(c-fe) ^\|z-a\|-r' when \|z - a\| is large enough the right-hand side is < 2n, and as j(z; y) is an integer, this implies y(^ 5 ?) = 0. But the exterior of D is connected, as seen above, hence the conclusion by (9.8.3).

(9.8.6) For any circuit y in C, defined in I, the set of points x e C - y(I) such thatj(x; y) ^ 0 is relatively compact in C. For by (9.8.5), that set is contained in any closed ball containing y(I).

(9.8.7) Let Aa C be a simply connected domain, y a circuit in A. For any point x ofC - A,y(*; y) = 0.